Micromechanical Multiscale Fracture Model for Compressive Strength of Blended Cement Pastes

Cement and Concrete Research 83 (2016) 188–202 Contents lists available at ScienceDirect Cement and Concrete Research journal homepage: www.elsevier.com/locate/cemconres Micromechanical multiscale fracture model for compressive strength of blended cement pastes Michal Hlobil a,b,⁎,VítŠmilauer a, Gilles Chanvillard c,1 a Department of Mechanics, Faculty of Civil Engineering, CTU in Prague, Thákurova 7, 166 29 Prague 6, Czech Republic b Institute for Mechanics of Materials and Structures (IMWS), Vienna University of Technology (TU Wien), Karlsplatz 13/202, A-1040 Wien, Austria c Lafarge Centre de Recherche, 95 Rue du Montmurier, 38290 Saint-Quentin Fallavier, France article info abstract Article history: The evolution of compressive strength belongs to the most fundamental properties of cement paste. Driven by an Received 19 June 2015 increasing demand for clinker substitution, the paper presents a new four-level micromechanical model for the Accepted 8 December 2015 prediction of compressive strength of blended cement pastes. The model assumes that the paste compressive Available online 9 March 2016 strength is governed by apparent tensile strength of the C-S-H globule. The multiscale model takes into account the volume fractions of relevant chemical phases and encompasses a spatial gradient of C-S-H between individual Keywords: grains. The presence of capillary pores, the C-S-H spatial gradient, clinker minerals, SCMs, other hydration prod- Compressive strength (C) Calcium-Silicate-Hydrate (C-S-H) (B) ucts, and air further decrease compressive strength. Calibration on 95 experimental compressive strength values Microstructure (B) shows that the apparent tensile strength of the C-S-H globule yields approx. 320 MPa. Sensitivity analysis reveals Cement paste (D) that the “C-S-H/space” ratio, followed by entrapped or entrained air and the spatial gradient of C-S-H, have the Blended cement (D) largest influence on compressive strength. © 2015 Elsevier Ltd. All rights reserved. 1. Introduction dealing with blended binders [5]. The main reason lies in altered chem- istry where a chemically heterogeneous gel differs in chemical phases, Compressive strength of concrete and its evolution belong to the e.g. CH may be depleted in pozzolanic reactions or C-A-H phases most important and tested parameters. Due to the multiscale nature may emerge. To overcome these deficiencies, micromechanical models of concrete spanning the range from sub-nanometers to meters and taking into account volume fractions directly have been set up for its composite nature, several factors on different scales play a role in more fundamental understanding. concrete compressive strength [1]. The most relevant initial factors in- Continuum micromechanical models able to reproduce the evolu- clude the binder type, aggregate type, extent of the interfacial transition tion of stiffness and compressive strength have recently been published, zone, and the air content. Time-dependent factors play a role mainly in e.g. for Portland-based materials [6,7] or for cocciopesto mortars [8]. binder’s reaction kinetics which is reflected in the evolution of chemical Pichler et al. [9] used spherical and acicular representation of hydrates phases with a direct impact on stiffness and strength evolution. to capture percolation threshold during hydration and the onset on Up to the time of Hoover Dam construction in the 1930s, a cementi- elasticity and strength. Hydrates consisted of solid C-S-H, small and tious binder was mostly equivalent to Portland cement. Since that time, large gel pores and other hydration products. The strength criterion Portland clinker has been substituted more and more by supplementary was based on deviatoric strength of hydrates which was identified to cementitious materials such as slag, fly ash, limestone or silica fume. The be 69.9 MPa. Once the quadratic stress average in arbitrarily oriented substitution rose from 17% in 1990 to 25% in 2010 among the top world needle-shaped hydrates exceeded this strength, the material failed in cement producers [2]. A further shift is expected, motivated by econom- a brittle manner. Similar modeling approach was also used for C3S, ical, ecological and sustainable benefits [3,4]. C2S, C3S+C3A + gypsum pastes, assuming that only C-S-H with the The question on the origin of concrete strength becomes reinitiated Mohr–Coulomb quasi-brittle failure criterion was responsible for the with the advent of blended binders. The famous Powers' empirical rela- compressive strength of pastes [10]. Computational micromechanical tionship between the gel-space ratio and compressive strength devel- models generally allow taking into account nonlinear elasto-plasto- oped for Portland-based materials needs several adjustments when damage constitutive laws at the expense of computational time. Several 2D and 3D lattice and continuum models were applied for cement paste [11] or concrete [12,13,14] to mention a few. ⁎ Corresponding author. In this paper, a new four-level micromechanical fracture model for E-mail address: [email protected] (M. Hlobil). 1 Gilles Chanvillard (21.1.1963 - 24.10.2015) passed away very unexpectedly during the blended cement pastes is presented, starting from C-S-H globule up to revision process of this paper. cement paste with entrapped/entrained air. The lowest homogenization http://dx.doi.org/10.1016/j.cemconres.2015.12.003 0008-8846/© 2015 Elsevier Ltd. All rights reserved. M. Hlobil et al. / Cement and Concrete Research 83 (2016) 188–202 189 level contains C-S-H globules, small and large gel pores [15]. A C-S-H strength, ft. Damage mechanics uses the concept of the equivalent globule is considered to be the only strain-softening component in the strain, εe, as a descriptor of damage evolution. Damage becomes initiated multiscale model, leading essentially to failure at each level. Softening when the equivalent strain, eε, exceeds strain at the onset of cracking, occurs under excessive tension or compression described by an elasto- ε0 =ft/E, where E is the elastic modulus. The Rankine criterion for tensile damage constitutive law introduced in Section 2.1. Higher levels also failure defines εe as use the elasto-damage constitutive law for a phase containing C-S-H σ globules with an updated homogenized stiffness, strength, and estimat- e 1 ε ¼ ; σ 1 N 0 ð1Þ ed fracture energy. Hence, the strength of cement paste origins from the E softening and failure of C-S-H globule. The multiscale model accounts where σ is the maximum positive effective principal stress on for volume fractions of C-S-H, other hydrates, capillary porosity, clinker, 1 undamaged-like material, see Fig. 1 (a). supplementary cementitious materials, entrained/entrapped air and Under uniaxial compressive stress, crack initiation occurs under a considers the spatial gradient of C-S-H among clinker grains. different mechanism. A homogeneous material experiences only one The model contains two independent variables that need to be cali- negative principal stress and the deviatoric stress. Cracking in diagonal, brated from experimental data: the apparent tensile strength of C-S-H shear band zone is often encountered on cementitious specimens, how- globules and the spatial gradient of C-S-H. Numerical results for elastic ever, the physical mechanism is again tensile microcracking in voids modulus and compressive/tensile strength on each scale are further and defects of the underlying microstructure [pp 297, 17]. Such a behav- fitted to microstructure-calibrated analytical expressions, speeding up ior has already been described in the work of Griffith [18], and McClin- the whole validation part. Sensitivity analysis identifies key compo- tock and Walsh [19], and we briefly review this theory and extend it nents for the compressive strength of blended pastes. with an equivalent strain to be used in the framework of damage mechanics. 2. Finite element analysis It is assumed that a material contains randomly oriented 2D elliptical flat voids with various aspect ratios m=b/a,seeFig. 1 (b). Further nota- 2.1. Fracture material model and its implementation tions assume that tensile stress is positive and σ1 ≥σ3.Thevoidshavea negligible area and only represent stress concentrators and internal de- The nonlinear constitutive behavior of quasi-brittle materials, such fects in a material. Under macroscopic biaxial stress, the maximum ten- as cement paste, mortar or concrete, can be generally described by sile stress among all voids, m⋅ση, appears on a critically inclined three common theories: plasticity, fracture mechanics, damage me- elliptical void under a critical angle ψ chanics, or their combinations. Plasticity fails to describe stiffness degradation which is needed for strain softening, strain localization, and the σ 3 À σ 1 σ 1 1 size effect, although several extensions were proposed [16]. Linear elas- cos2ψ ¼ ; ≥ À ð2Þ 2ðÞσ 3 þ σ 1 σ 3 3 tic fracture mechanics can only deal with brittle materials with a negligible process zone ahead of a crack tip and cannot handle microcrack 2 ÀðÞσ 1 À σ 3 nucleation into a macrocrack. Damage mechanics, particularly the cohe- m Á σ η ¼ ð3Þ 4ðÞσ þ σ sive crack model, defines the traction–separation law in a plastic zone of 1 3 an opening crack which is related to the fracture energy of a material Crack formation occurs when the tangential tensile stress, m⋅ση,equals and stiffness degradation. This nonlinear fracture mechanics approach to the tensile strength of the matrix. Since ση and the crack geometry, m, offers reasonable stress–strain predictions with minimum parameters, cannot be measured directly, it is reasonable to relate their product to reasonable computation time, and captures the size effect of quasibrittle the uniaxial macroscopic tensile stress, σ , as proposed by Griffith [18] materials [17]. More elaborated plastic-damage models have been suc- 1 cessfully used for the mesoscale analysis of concrete [14]. m Á σ η σ ¼ ð4Þ The present material model is based on fracture mechanics on all 1 2 considered four scales.

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